How to clean up intersecting faces and edges?
Edit: As your request has changed I give a second answer below.

Maybe you're not choosing the best way to build your object, in your case the Solidify modifier will create unwanted intersection and it will be hard to fix.

What you could do for example is begin with a triangle (3 vertices circle), extrude it up, then select the 3 faces, press AltE to extrude, choose Individual Faces and drag to extrude.



Second answer:


Create a 10 vertices circle, scale one of 2 vertices so that it makes a hexagon.





Move down those vertices, extrude down the whole mesh.





Select one of 2 edges underneath, bevel them, extrude down.